Triangle calculator SSA

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Triangle has two solutions with side c=9.6443904215 and with side c=1.84767624318

#1 Acute scalene triangle.

Sides: a = 7.5   b = 6.2   c = 9.6443904215

Area: T = 23.2466183019
Perimeter: p = 23.3443904215
Semiperimeter: s = 11.67219521075

Angle ∠ A = α = 51.0388226456° = 51°2'18″ = 0.8910785096 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 88.9621773544° = 88°57'42″ = 1.55326758568 rad

Height: ha = 6.19989821384
Height: hb = 7.49987687158
Height: hc = 4.82109070726

Median: ma = 7.18774852524
Median: mb = 8.06333395224
Median: mc = 4.90985413183

Inradius: r = 1.99216276905
Circumradius: R = 4.82327438633

Vertex coordinates: A[9.6443904215; 0] B[0; 0] C[5.74553333234; 4.82109070726]
Centroid: CG[5.13297458461; 1.60769690242]
Coordinates of the circumscribed circle: U[4.82219521075; 0.08773856033]
Coordinates of the inscribed circle: I[5.47219521075; 1.99216276905]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.9621773544° = 128°57'42″ = 0.8910785096 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 91.0388226456° = 91°2'18″ = 1.55326758568 rad




How did we calculate this triangle?

1. Use Law of Cosines

a = 7.5 ; ; b = 6.2 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6.2**2 = 7.5**2 + c**2 -2 * 7.5 * c * cos (40° ) ; ; ; ; c**2 -11.491c +17.81 =0 ; ; p=1; q=-11.491; r=17.81 ; ; D = q**2 - 4pr = 11.491**2 - 4 * 1 * 17.81 = 60.7954199875 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.49 ± sqrt{ 60.8 } }{ 2 } ; ; c_{1,2} = 5.74533332 ± 3.89857089161 ; ; c_{1} = 9.64390421161 ; ;
c_{2} = 1.84676242839 ; ; ; ; text{ Factored form: } ; ; (c -9.64390421161) (c -1.84676242839) = 0 ; ; ; ; c>0 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 7.5 ; ; b = 6.2 ; ; c = 9.64 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 7.5+6.2+9.64 = 23.34 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.34 }{ 2 } = 11.67 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.67 * (11.67-7.5)(11.67-6.2)(11.67-9.64) } ; ; T = sqrt{ 540.39 } = 23.25 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23.25 }{ 7.5 } = 6.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23.25 }{ 6.2 } = 7.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23.25 }{ 9.64 } = 4.82 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 6.2**2+9.64**2-7.5**2 }{ 2 * 6.2 * 9.64 } ) = 51° 2'18" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.5**2+9.64**2-6.2**2 }{ 2 * 7.5 * 9.64 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 51° 2'18" - 40° = 88° 57'42" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23.25 }{ 11.67 } = 1.99 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.5 }{ 2 * sin 51° 2'18" } = 4.82 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.2**2+2 * 9.64**2 - 7.5**2 } }{ 2 } = 7.187 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.64**2+2 * 7.5**2 - 6.2**2 } }{ 2 } = 8.063 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.2**2+2 * 7.5**2 - 9.64**2 } }{ 2 } = 4.909 ; ;







#2 Obtuse scalene triangle.

Sides: a = 7.5   b = 6.2   c = 1.84767624318

Area: T = 4.45215350344
Perimeter: p = 15.54767624318
Semiperimeter: s = 7.77333812159

Angle ∠ A = α = 128.9621773544° = 128°57'42″ = 2.25108075576 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 11.0388226456° = 11°2'18″ = 0.19326533952 rad

Height: ha = 1.18770760092
Height: hb = 1.43659790434
Height: hc = 4.82109070726

Median: ma = 2.62196880997
Median: mb = 4.49766949796
Median: mc = 6.81985311564

Inradius: r = 0.57326639297
Circumradius: R = 4.82327438633

Vertex coordinates: A[1.84767624318; 0] B[0; 0] C[5.74553333234; 4.82109070726]
Centroid: CG[2.53106985851; 1.60769690242]
Coordinates of the circumscribed circle: U[0.92333812159; 4.73435214694]
Coordinates of the inscribed circle: I[1.57333812159; 0.57326639297]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 51.0388226456° = 51°2'18″ = 2.25108075576 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 168.9621773544° = 168°57'42″ = 0.19326533952 rad

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How did we calculate this triangle?

1. Use Law of Cosines

a = 7.5 ; ; b = 6.2 ; ; beta = 40° ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 6.2**2 = 7.5**2 + c**2 -2 * 7.5 * c * cos (40° ) ; ; ; ; c**2 -11.491c +17.81 =0 ; ; p=1; q=-11.491; r=17.81 ; ; D = q**2 - 4pr = 11.491**2 - 4 * 1 * 17.81 = 60.7954199875 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.49 ± sqrt{ 60.8 } }{ 2 } ; ; c_{1,2} = 5.74533332 ± 3.89857089161 ; ; c_{1} = 9.64390421161 ; ; : Nr. 1
c_{2} = 1.84676242839 ; ; ; ; text{ Factored form: } ; ; (c -9.64390421161) (c -1.84676242839) = 0 ; ; ; ; c>0 ; ; : Nr. 1
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 7.5 ; ; b = 6.2 ; ; c = 1.85 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 7.5+6.2+1.85 = 15.55 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.55 }{ 2 } = 7.77 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.77 * (7.77-7.5)(7.77-6.2)(7.77-1.85) } ; ; T = sqrt{ 19.82 } = 4.45 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.45 }{ 7.5 } = 1.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.45 }{ 6.2 } = 1.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.45 }{ 1.85 } = 4.82 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 6.2**2+1.85**2-7.5**2 }{ 2 * 6.2 * 1.85 } ) = 128° 57'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.5**2+1.85**2-6.2**2 }{ 2 * 7.5 * 1.85 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 128° 57'42" - 40° = 11° 2'18" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.45 }{ 7.77 } = 0.57 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.5 }{ 2 * sin 128° 57'42" } = 4.82 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.2**2+2 * 1.85**2 - 7.5**2 } }{ 2 } = 2.62 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.85**2+2 * 7.5**2 - 6.2**2 } }{ 2 } = 4.497 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.2**2+2 * 7.5**2 - 1.85**2 } }{ 2 } = 6.819 ; ;
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